Entity-oriented spatial coding scheme and its application for spatial topology

Figures & data

Figure 1. A full-OACD generated from 2 entities.

Figure 2. The procedure of generating a full-OACD(3, R²). (a) the generator set consists of three points marked with color R, G, and B; (b)-(d) Half-plane partitions and assignments of chromatic codes with respect to perpendicular bisectors pb<b, G>, pb<g, R>, and pb<r, B>, respectively. (e) Overlapping all the six half-planes in (b)-(d) together; and (f) Adding all chromatic components together to form the chromatic codes.

Figure 3. Two examples of full-OACDs. (a) a full-OACD(4, R²); (b) the homomorphic part of a full-OACD(6, R²).

Figure 4. Two basic structural units of full-OACD. (a) 2-I unit; (b) 3-I unit.

Figure 5. Three types of particle relations in 2-I/3-I units: adjacent (Adj.), interval (Int.) and opposite (Opp.).

Table 1. Chromatic and code distances between two particles in 2-I/3-I units.

			2-I Units			3-I Units
Ω1	Ω2	Relation	δ	γ	Base	δ	γ	Base
Vertex φ	Edge η	Adjacent	1	2	$\mathit{/}\cong$	2	3	$\mathit{/}\cong$
Vertex φ	Cell ζ	Adjacent	2	4	$\mathit{/}\cong$	2	2	$\mathit{/}\cong$
Edge η1	Edge η2	Adjacent	2	4	$\mathit{/}\cong$	2	3	$\mathit{/}\cong$
Edge η1	Edge η2	Opposite	2	2	$\cong$	4	3	$\mathit{/}\cong$
Edge η1	Edge η2	Interval	n/a	n/a	n/a	3	2	$\cong$
Edge η	Cell ζ	Adjacent	1	2	$\mathit{/}\cong$	1	2	$\mathit{/}\cong$
Edge η	Cell ζ	Opposite	3	4	$\mathit{/}\cong$	4	3	$\mathit{/}\cong$
Edge η	Cell ζ	Interval	n/a	n/a	n/a	3	3	$\mathit{/}\cong$
Cell ζ1	Cell ζ2	Adjacent	2	2	$\cong$	2	2	$\cong$
Cell ζ1	Cell ζ2	Opposite	4	4	$\cong$	4	2	$\cong$
Cell ζ1	Cell ζ2	Interval	n/a	n/a	n/a	4	3	$\cong$

Note: The spatial relations listed in this table are shown in Figure 7.

Figure 6. General spatial topological relations among particles in full-OACDs.

Figure 7. Six types of complex topological relations in full-OACDs.

Figure 8. Other types of spatial chromatic tessellations. (a) Diagrams generated by weighted bisectors; (b) Four generators are concyclic.

Data availability statement

Data sharing is not applicable to this article as no new data were created or analyzed in this study.